#### Abstract

We present the necessary and sufficient condition for the monotonicity of the ratio of the power and second Seiffert means. As applications, we get the sharp upper and lower bounds for the second Seiffert mean in terms of the power mean.

#### 1. Introduction

Throughout this paper, we assume that with . The second Seiffert mean and th power mean of and are defined by respectively.

It is well-known that the power mean is strictly increasing with respect to for fixed with . In the recent past, both mean values have been the subject of intensive research. In particular, many remarkable inequalities for and can be found in the literature [1–5].

Seiffert [6] proved that the double inequality holds for all with .

In [7], Hästö proved that the function is strictly increasing on if and presented an improvement for the first inequality in (3).

Costin and Toader [8] proved that the inequality holds for all with .

In [9], Witkowski proved that the double inequality holds for all with .

Recently, the following optimal estimations for the second Seiffert mean by power means were obtained independently in [10, 11]: for all with .

The main purpose of this paper is to give the necessary and sufficient condition for the monotonicity of the function on and present the best possible parameters and such that the double inequality holds for all with .

#### 2. Main Results

In order to prove our main results we first establish a lemma.

Lemma 1. *Let be defined on by
**
Then there exists such that is strictly decreasing with respect to on and strictly increasing with respect to on if .*

*Proof. *Let
Then,

We divide two cases to prove that for all and .

*Case 1. *Consider that . From (14) we clearly see that

Equation (15) implies that is strictly concave with respect to on the interval . Then (16) and the basic properties of concave function lead to the conclusion that

*Case 2. *Consider that . Making use of the weighted arithmetic-geometric inequality we get

Equations (14) and (18) lead to

Note that

It follows from (20) and the concavity of the function with respect to on the interval that
Therefore, follows from (19) and (21).

Next we prove the desired result. From (12) and (13) together with the fact that we clearly see that there exists such that is strictly decreasing with respect to on and strictly increasing with respect to on . Therefore, Lemma 1 follows easily from (10) and (11) together with the piecewise monotonicity of with respect to on the interval .

Theorem 2. *Let be defined on by
**
Then the following statements are true.*(1)* is strictly increasing with respect to on if and only if .*(2)* is strictly decreasing with respect to on if and only if .*(3)*If , then there exists such that is strictly increasing with respect to on and strictly decreasing with respect to on .*

*Proof. *It follows from (22) and (23) that
where is defined by (8). And
where

(1) If is strictly increasing with respect to on , then (24) leads to for all . Making use of L’Höspital’s rule and (8) we get
which implies that .

If , then from (8) and (26) together with the fact that the function is strictly increasing on we get
for all .

Equations (8) and (25) together with inequality (29) lead to the conclusion that
for all .

Therefore, is strictly increasing with respect to on which follows easily from (24), (28), and (30).

(2) If is strictly decreasing with respect to on , then (24) implies that for all . In particular, we have and . Indeed, if , then (8) leads to the conclusion that .

If , then from (8) and (26) together with the fact that the function is strictly increasing on we get
for all .

Equations (8) and (25) together with inequality (32) lead to the conclusion that
for all .

Therefore, is strictly decreasing with respect to on which follows easily from (24), (31), and (33).

(3) If , then (8) leads to

It follows from Lemma 1 and (34) that we clearly see that there exists such that for and for . Then from (24) we get Theorem 2(3) immediately.

Theorem 3. *For all with , the double inequality
**
holds with the best possible constants and , where is the solution of the equation on and and are defined by (8) and (22), respectively.*

*Proof. *Without loss of generality, we assume that . Let ; then from (1) and (2) we get

If , then from (22) and Theorem 2(1) we get

Therefore, the first inequality in (35) with the best possible constant follows from (36) and (37) together with the monotonicity of given in Theorem 2(1).

If , then Lemma 1 and (24) together with (34) imply that there exists such that , and is strictly increasing on and strictly decreasing on . Therefore, we have

Making use of MATHEMATICA software, numerical computations show that

Therefore, the second inequality in (35) with the best possible constant follows from (36) and (38) together with the piecewise monotonicity of .

Corollary 4. *The double inequality
**
holds for all with if and only if and , where and are, respectively, the quadratic and th Lehmer means of and .*

*Proof. *Without loss of generality, we assume that . Let . Then from Theorem 2 and (24) we clearly see that the if and only if and if and only if . Then (8) leads to the conclusion that the inequalities
hold for all if and only and .

Therefore, Corollary 4 follows easily from inequalities (41) and (42) together with (1).

Corollary 5. *Let with . Then Theorem 2 leads to the following Ky Fan type inequality:
**
if .*

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This research was supported by the Natural Science Foundation of China under Grants 61374086, 11371125, and 11171307, the Natural Science Foundation of Hunan Province under Grant 12C0577, and the Natural Science Foundation of Zhejiang Province under Grant LY13A010004.